mtl file after exporting comes without texture map (map_kd ...)
The problem is that the .obj exporter requires a material setup using the Principled Shader in order to write an .mtl with the map_Kd attribute. Unfortunately the documentation is still work in progress for Blender 2.8, that's why this wasn't previously documented.

The fix for your problem is to replace the Diffuse BSDF nodes with the Principled BSDF. Then you can simply plug your texture into the base color socket of the Principled BSDF, set Specular to zero and it should export as expected.